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RL-RC Circuits
&
Applications
SVES Circuits Theory
Introduction
• In this chapter, phasor algebra will be used to
develop a quick, direct method for solving both
the series and the parallel ac circuits.
• Describe the relationship between current and
voltage in an RC & RL circuits
• Determine impedance and phase angle in RC
and RL circuits
Impedance and the Phasor Diagram
• Resistive Elements
• Use R=0° in the following polar format to ensure the
proper phase relationship between the voltage and the
current resistance:
• The boldface Roman quantity ZR, having both
magnitude and an associate angle, is referred to as the
impedance of a resistive element.
• ZR is not a phasor since it does not vary with time.
• Even though the format R0° is very similar to the phasor
notation for sinusoidal current and voltage, R and its
associated angle of 0° are fixed, non-varying quantities.
Resistive ac circuit.
Resistive ac circuit
Voltage is 100 volts Peak
Waveforms for Last Example
Resistive
Phasor diagram of Example Resistive
20.0
100
Analysis of Resistive Circuits
• The application of Ohm’s law to series circuits
involves the use of the quantities Z, V, and I as:
V = IZ
I = V/Z
Z = V/I
R=Z
Impedance and the Phasor Diagram
• Capacitive Reactance (XC)
• Use C = – 90° in the following polar format for
capacitive reactance to ensure the proper phase
relationship between the voltage and current of an
capacitor:
• The boldface roman quantity Zc, having both
magnitude and an associated angle, is referred to as
the impedance of a capacitive element.
Impedance and the Phasor Diagram
• ZC is measured in ohms and is a measure of how
much the capacitive element will “control or impede”
the level of current through the network.
• This format like the one for the resistive element, will
prove to be a useful “tool” in the analysis of ac
networks.
• Be aware that ZC is not a phasor quantity for the same
reason indicated for a resistive element.
Analysis of Capacitive ac Circuit
• The current leads the
voltage by 90 in a
purely capacitive ac
circuit
Capacitive ac circuit.
Capacitive ac circuit, Voltage is 15 volts peak
Waveforms for Example
current leads the voltage by 90 degrees
Phasor diagrams for Example
7.50
15.00
Impedance and the Phasor Diagram
• Inductive Reactance (XL)
• Use L = 90° in the following polar format for
inductive reactance to ensure the proper phase
relationship between the voltage and the current of an
inductor:
• The boldface roman quantity ZL, having both
magnitude and an associated angle, in referred to as
the impedance of an inductive element.
Impedance and the Phasor Diagram
• ZL is measured in ohms and is a measure of how much
the inductive element will “control or impede” the level
of current through the network.
• This format like the one for the resistive element, will
prove to be a useful “tool” in the analysis of ac
networks.
• Be aware that ZL is not a phasor quantity for the same
reason indicated for a resistive element.
Inductive ac circuit.
Inductive ac circuit Voltage is 24 volts Peak
Inductor Waveforms for Example
voltage leads the current by 90 degrees
Phasor diagrams for Example.
24.0 V
8.0
Three cases of impedance
R – C series circuit
Illustration of sinusoidal response with general phase relationships
of VR, VC, and I relative to the source voltage. VR and I are in the
phase; VR leads VS; VC lags VS; and VR and VC are 90º out of phase.
Impedance of a series RC circuit.
Development of the impedance triangle
for a series RC circuit.
Impedance of a series RC circuit.
Impedance of a series RC circuit.
Phase relation of the voltages and current in
a series RC circuit.
Voltage and current phasor diagram for the waveforms
Voltage diagram for the voltage in a R-C circuit
Voltage diagram for the voltage in a R-C circuit
An illustration of how Z and XC change with frequency.
As the frequency increases, XC decreases, Z decreases, and
 decreases. Each value of frequency can be visualized as
forming a different impedance triangle.
Illustration of sinusoidal response with general phase relationships
of VR, VL, and I relative to the source voltage. VR and I are in phase;
VR lags VS; and VL leads VS. VR and VL are 90º out of phase with
each other.
Impedance of a series RL circuit.
Development of the Impedance triangle
for a series RL circuit.
Impedance of a series RL circuit.
Impedance of a series RL circuit.
Phase relation of current and voltages in a series RL circuit.
Voltage phasor diagram for the waveforms .
Voltage and current phasor diagram for the waveforms
Voltages of a series RL circuit.
61 V
Voltages of a series RL circuit.
Reviewing the frequency response of the
basic elements.
Frequency Selectivity of RC
Circuits
• Frequency-selective circuits permit signals of
certain frequencies to pass from the input to the
output, while blocking all others
• A low-pass circuit is realized by taking the output
across the capacitor, just as in a lag network
• A high-pass circuit is implemented by taking the
output across the resistor, as in a lead network
The RC lag network (Vout = VC)
FIGURE 10-17 An illustration of how Z and XC change with frequency.
Frequency Selectivity of RC
Circuits
• The frequency at
which the capacitive
reactance equals the
resistance in a lowpass or high-pass RC
circuit is called the
cutoff frequency:
fc = 1/(2RC)
Normalized general response curve of a low-pass
RC circuit showing the cutoff frequency and the
bandwidth
- 3 dB point
normalized
Cutoff point
Example of low-pass filtering action. As
frequency increases, Vout decreases
The RC lead network (Vout = VR)
Example of high-pass filtering action. As
frequency increases, Vout increases
High-pass filter responses.
High-pass filter responses, filters in series
Each r-c combination
- 20dB / decade
Observing changes in Z and XL with frequency by
watching the meters and recalling Ohm’s law
RL Circuit as a Low-Pass Filter
• An inductor acts as a short to dc
• As the frequency is increased, so does the
inductive reactance
– As inductive reactance increases, the output
voltage across the resistor decreases
– A series RL circuit, where output is taken
across the resistor, finds application as a lowpass filter
Example of low-pass filtering action. Winding resistance
has been neglected. As the input frequency increases, the
output voltage decreases
RL Circuit as a High-Pass Filter
• For the case when output voltage is
measured across the inductor
– At dc, the inductor acts a short, so the output
voltage is zero
– As frequency increases, so does inductive
reactance, resulting in more voltage being
dropped across the inductor
– The result is a high-pass filter
FIGURE 12-39 Example of high-pass filtering action. Winding resistance has been neglected. As the input frequency increases, the
output voltage increases.

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